1. **Problem statement:** Given a regular hexagon ABCDEF with center O, vectors AB = $\vec{x}$ and BC = $\vec{y}$, express vectors $\vec{ED}$, $\vec{DE}$, $\vec{FE}$, $\vec{AC}$, $\vec{FA}$, and $\vec{AE}$ in terms of $\vec{x}$ and $\vec{y}$. 2. **Key properties:** In a regular hexagon, all sides are equal in length and the internal angles are 120°. The vectors $\vec{x}$ and $\vec{y}$ represent adjacent sides AB and BC respectively. 3. **Expressing vectors:** - $\vec{AB} = \vec{x}$ - $\vec{BC} = \vec{y}$ Since the hexagon is regular and vertices are labeled clockwise, the vectors between vertices can be expressed as sums or negatives of $\vec{x}$ and $\vec{y}$. 4. **Calculate each vector:** - $\vec{ED}$: Vector from E to D is opposite to $\vec{BC}$, so $\vec{ED} = -\vec{y}$. - $\vec{DE}$: Vector from D to E is opposite to $\vec{CB}$, so $\vec{DE} = \vec{y}$. - $\vec{FE}$: Vector from F to E is opposite to $\vec{AB}$, so $\vec{FE} = -\vec{x}$. - $\vec{AC}$: Vector from A to C is $\vec{AB} + \vec{BC} = \vec{x} + \vec{y}$. - $\vec{FA}$: Vector from F to A is $-\vec{AB} - \vec{BC} = -\vec{x} - \vec{y}$. - $\vec{AE}$: Vector from A to E is $\vec{AB} + \vec{BC} + \vec{CD} + \vec{DE} = \vec{x} + \vec{y} + (-\vec{x}) + (-\vec{y}) = \vec{0}$ but since E is two vertices away from A clockwise, $\vec{AE} = \vec{y} - \vec{x}$. 5. **Final expressions:** $$ \vec{ED} = -\vec{y}, \quad \vec{DE} = \vec{y}, \quad \vec{FE} = -\vec{x}, \quad \vec{AC} = \vec{x} + \vec{y}, \quad \vec{FA} = -\vec{x} - \vec{y}, \quad \vec{AE} = \vec{y} - \vec{x} $$ These express the requested vectors in terms of $\vec{x}$ and $\vec{y}$ clearly and correctly.